Controllability of a viscoelastic plate using one boundary control in displacement or bending

In this paper we consider a viscoelastic plate (linear viscoelasticity of the Maxwell-Boltzmann type) and we compare its controllability properties with the (known) controllability of a purely elastic plate (the control acts on the boundary displacement or bending). By combining operator and moment methods, we prove that the viscoelastic plate inherits the controllability properties of the purely elastic plate

Well-Posedness and Polynomial energy decay rate of a transmission problem for Rayleigh beam model with heat conduction

In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem's well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt-Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams

Twenty years of distributed port-Hamiltonian systems:A literature review

The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups

Stabilization of a Quantum Equation under Boundary Connections with an Elastic Wave Equation

The stability of coupled PDE systems is one of the most important topic because it covers realistic modeling of the most important physical phenomena. In fact, the stabilization of the energy of partial differential equations has been the main goal in solving many structural or microstructural dynamics problems. In this chapter, we investigate the stability of the Schrödinger-like quantum equation in interaction with the mechanical wave equation caused by the vibration of the Euler–Bernoulli beam, to effect stabilization, viscoelastic Kelvin-Voigt dampers are used through weak boundary connection. Firstly, we show that the system is well-posed via the semigroup approach. Then with spectral analysis, it is shown that the system operator of the closed-loop system is not of compact resolvent and the spectrum consists of three branches. Finally, the Riesz basis property and exponential stability of the system are concluded via comparison method in the Riesz basis approach

A unified framework for approximation in inverse problems for distributed parameter systems

A theoretical framework is presented that can be used to treat approximation techniques for very general classes of parameter estimation problems involving distributed systems that are either first or second order in time. Using the approach developed, one can obtain both convergence and stability (continuous dependence of parameter estimates with respect to the observations) under very weak regularity and compactness assumptions on the set of admissible parameters. This unified theory can be used for many problems found in the recent literature and in many cases offers significant improvements to existing results

Linear-Quadratic Control of a MEMS Micromirror Using Kalman Filtering

The deflection limitations of electrostatic flexure-beam actuators are well known. Specifically, as the beam is actuated and the gap traversed, the restoring force necessary for equilibrium increases proportionally with the displacement to first order, while the electrostatic actuating force increases with the inverse square of the gap. Equilibrium, and thus stable open-loop voltage control, ceases at one-third the total gap distance, leading to actuator snap-in. A Kalman Filter is designed with an appropriately complex state dynamics model to accurately estimate actuator deflection given voltage input and capacitance measurements, which are then used by a Linear Quadratic controller to generate a closed-loop voltage control signal. The constraints of the latter are designed to maximize stable control over the entire gap. The design and simulation of the Kalman Filter and controller are presented and discussed, with static and dynamic responses analyzed, as applied to basic, 100 micrometer by 100 micrometer square, flexure-beam-actuated micromirrors fabricated by PolyMUMPs. Successful application of these techniques enables demonstration of smooth, stable deflections of 50% and 75% of the gap